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In "Physics and Chemistry of the Solar System" Jeans' Criterion is given as:

$\frac{GmM}{R_c} = \frac{3mkT}{2}$

... To me this suggests that on the left we have Joules, and on the right we have kg$\cdot$joules.

I then went to some old lecture notes and found that the professor derived the Jean's Radius from:

$\frac{GM}{R_c} = \frac{3kT}{2}$

which also has (as far as I can tell) an issue with units. Can anyone help my understand what I'm missing with this equation?

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Your first and second equations are the same with a factor of $m$ removed from both sides. Are the $m$ on the left and right side different?

The derivation in Wikipedia ends up with:

$$ kT = \frac{GM\mu}{r} $$

where the left side is the energy per particle and on the right side $\mu$ is the mass per particle, and this is dimensionally consistent. This is basically the same as your second equation if the $\mu$ is added.

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Often "mu" is written in a dimensionless way, so that the mean mass per particle m = mu * amu, where amu is the mass in amu. It seems likely that your "m" factors are actually mass per particles. –  user2153903 Mar 10 '13 at 14:16
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